Saturday, 16 February 2013

Idea on Genetic Algorithm

In the mid 1800's a British naturalist named Charles Darwin published a book named 'The Origin of Species' and said that including human all the living being were not put into the earth by God, rather we evolved from other species. At that time it sounded absurd. But with the time flow when we invented more and more new stuffs then we gradually understood that what Darwin said was right. Overtime every living being is evolving to keep up with the changing environment. If it can evolve then it will survive otherwise nature will make it vanish.

A] Background:-

Now let us concentrate on what Evolution is and how every living being is evolving. Evolution is the process of change in the inherited traits from parents over generations. Every living being gets some traits(encoded in chromosome) from its parent and it make these traits flow over its generation. During these flow some chromosome combines with other type of chromosome and new types of chromosome with new traits that belongs to neither of the participating chromosome are evolved. This new type of chromosome may be more superior or inferior.

B]Why Genetic Algorithms:-
First let me tell you something about “hard computing” and “soft computing”. Hard computing is the conventional computing that requires a precisely stated analytical model and returns deterministic output. For example ...
But soft computing is unlike hard computing, it is tolerant of imprecision, uncertainty, partial truth, and
approximation. In soft computing algorithm learns to give solution. For example Handwritten character recognition .
Genetic algorithm(GA) is also a tool of soft computing. GA is one of the best way to solve the type of the problem about which very little is known. All you need to know what you need the solution to be able to do well and GA will provide you the solution. GA is basically a search algorithm that are applied to search for solution in a large state space that is uneven in nature.
GA is useful when---
=>The search space is large and complex.
=>No mathematical analysis is available.
=>The problem is to find solution under multi-objective function.
=>Conventional method fails.

C] Basic Principles:-
GA is based on the theory of natural selection and work on generating a set of random solutions and making them compete in an arena where only the fittest survives.
In GA we initially generate a fixed number of chromosome that will form the Initial Population. Actually each chromosome represents a possible solution and you will ask how? This actually depends upon the programmer, how he maps the actual problem to GA space. We need to perform this mapping so that we can use GA operators on these chromosomes.
For example let a maximization problem where f(x)=x^2 and x is a integer that belongs to [0,31]. Here we can represent the possible values i.e integer decimal values into string of binary numbers of length 5. Our Initial population may contain chromosomes 10001(17),01010(10),11001(26),10100(20).
Then we will select chromosomes with some selection procedure that will go to matting pool where all GA operators will be applied . In matting pool chromosomes are selected in pairs and crossover operator is applied on them and similarly mutation operator is also applied on them. This process will continue until a termination condition arises. After the completion of the algorithm there is a high chance that we will get a feasible solution. Yes there is also a fitness measuring function that will keep track of the fitness of each chromosome, at the time of crossover we will replace some chromosome with least fitness, following the theory of 'Survival of the fittest'.Actually generally selection process and termination condition is dependent on this Fitness function. Thus starting from a set of random solutions the algorithm uses these operators and fitness function to guide its search for the optimal solution.

D] Methodology:-
1]Initialization:- To start GA first we need to define the Maximum allowable population(number of maximum chromosome), Maximum number of generation we will allow the algorithm to generate . We also need to define the crossover rate and mutation rate these two value will specify whether a particular chromosome will undergo crossover and mutation. Above all we need to generate the initial chromosomes with which we will start our selection procedure. We can apply some logic to assign the values of the chromosomes or simply can randomly assign values to chromosomes.
For example for above described problem we can generate chromosome using only two constraint one is that chromosome will contain only binary values and chromosome will be of length 5. Then we can generate chromosomes of length 5 up to Number of initial population.

2]Calculate Fitness: GA generates solution based on the fitness value of a chromosome. So after initialization we need to calculate the fitness value of each of the chromosome in the population. We will calculate the fitness value using a Objective function that is called a Fitness function.
2.1] Fitness function: A fitness function is a particular type of objective function that is used to evaluate the quality of a chromosome. It assigns a fitness value to all the chromosome in the population. This function will vary according to the problem. Fitness function should contain all the variables that affects the solution. For some constraint satisfaction problem there will be some constraints that will lead the search. Of course these constraint will have different effect on the solution. Some will have much effect and other will have less. So we need to assign some sort of weight to these constraint according to their effect on the solution. For example ,in a constraint satisfaction problem we can define a fitness function as f(X)= ((CS*X)-((N-X)*CNS))/(CS*N) where CS is cost for satisfying a constraint and CNS is cost for not satisfying a constraint. N is the total constraint applied on the problem and X is the number of satisfied constraint.
3]Selection:- GA use the selection operator to select chromosome from the population that will go to matting pool and various GA operator will be applied on them. As the selected chromosomes are the ones whose genes are inherited by the next generation, it is desirable that the selected chromosomes will have good fitness value. The convergence rate of a GA is highly determined by the selection method, with better selection process resulting in higher convergence rates. However, if the selection process is not good enough, the convergence rate will be slow. If the selection process is too active then it may create problem by converging to an sub-optimal solution.
3.1] Roulette Wheel Selection:- Roulette Wheel selection 
This can be simulated by following algorithm.
1.Calculate sum of all chromosome fitness(S) in population.
2.Generate random number (r) from interval (0,S).
3.Go through the population and sum fitness from first chromosome. When the sum is greater than r , stop and return the chromosome where you are.
3.2] Tournament selection:- It is also one of the popular method for selection procedure. In this method we first set a value for the size of tournament let it be k. Then we select k chromosomes randomly from the population. Now we have (n/k) number of sets each containing k chromosomes. Now select the chromosome with best fitness value from each set. Thus we can select n/k number of chromosomes which will go to matting pool.
For example let the population contains 4 chromosome as
chromosome no. Fitness value
1                                   2
2                                   1
3                                   4
4                                   3
Now let k=2 and we combine chromosome 1 and 2 in a set and 3 and 4 in other. From first set chromosome 1 will be selected and from second set chromosome 3 will be selected. Total 4/2=2 chromosome will be selected for matting pool.
4]Crossover:- It is performed on selected chromosomes to generate offspring. The idea behind crossover is that the offspring may inherit the good traits from its parent and a better chromosome will evolve. Thus GA offers an opportunity to share better solution from one to another.
First we need to find the chromosomes from matting pool that will undergo crossover using crossover rate. Crossover rate specifies the number of chromosomes that will undergo the crossover operation. It will be in the range [0,1]. To select chromosome from matting pool we will generate a random number “r” with value [0,1], if “r” is less than the Crossover rate then that chromosome will undergo crossover. Thus total chromosome that will undergo crossover operation is Total population * Crossover rate.
Then Crossover operator operates on the selected chromosomes in the following way..
[1]Choose a random point on the two parents
[2]Split parents at this crossover point
[3]Create children by exchanging tails
Crossover may be of various type like
Single Point :- In this case we randomly select a crossover point then swaps the contents of the parent chromosomes. For example, let crossover point be 5
Parent chromosome       Child chromosome
1278734919                   1278747933
8911947933                     8911934919
Two Point:- In this case we randomly selects two crossover point and then swaps the contents of the parent chromosomes. For the below example let the crossover points be 3 and 7
Parent chromosome                Child chromosome
1278734919                            1271947919
8911947933                              8918734933

Uniform Point:- bits are copied from the first or from the second parent depending upon the mask value ( 0 and 1 )that is generated randomly equal to the number of bits in the chromosome. For example let the mask be 1100100110 (let 0 means copy from first parent to first child and 0 means copy from second parent to first child).
Parent chromosome                 Child chromosome
1278734919                             1211747913
8911947933                               8978934939

5] Mutation :- During the execution algorithm may stuck into a local optima that is obviously not our required solution. To get rid of these situation we can use the mutation operator. It is the operator that operated upon the selected chromosome. Mutation randomly selects some mutation points and inverts the contents of those points. Thus mutation enables GA to jump to a random area in the solution space. The positions where mutation is taking place is called Mutation point. We will use Mutation operator to select the bits of chromosome where the mutation will take place. We can write algorithm for mutation operator as:
  1. For each chromosome (C) in matting pool
  2. For each bit in (C)
  3. generate a random number “r” in[0,1]
  4. if r<MUTATION_RATE then invert the bit
Here the total number of bits that will undergo mutation is equals to Number of chromosomes in matting pool * chromosome length * MUTATION RATE .
For example let there are two chromosomes (each of them of length 5) in matting pool and let the MUTATION_RATE=0.4
10101 and 01010. Now generate (5*2)=10 random numbers in[0,1] say, 0.1,0.5,0.3,0,7,0,3(for first chromosome ) and 0.6,0,35,0,24,0.9,0.12 (for second chromosome )
so for first chromosome 2nd and 4th bits will be inverted and for second chromosome 1st and 4th bits will be inverted. So the resultant chromosomes are: 11111 and 11000.

6] Termination:- Evolution in the real world of nature , is a continuous process. In the world of computing “how long do we need to continue the GA to get a feasible solution?” is a problem to be sorted out. In most of the problem we do not know the best value of the fitness function. We therefore stop the search when we feel that there has not been an improvement in either the maximum or average fitness of the population over several generations. The GA is then said to have done its work though here is never a guarantee that the global optimum has been reached.
The algorithm may take very long time to converge to a particular result where it may provide a solution that is acceptable. In that case algorithm will not stop unless we forcefully stop it. In that case we need to define the termination condition other than getting a actual solution like Acceptable error. Maximum allowable population may also be a termination criteria as we GA sometime requires an enormous amount of memory.

E]Flowchart:- We can represent the whole idea of GA in a flowchart form as:

Some of the fields where GA can be used are

DOMAINS                                      APPLICATION
Machine learning            designing neural networks, classification algorithms
Signal Processing            filter design
Scheduling                       manufacturing, resource allocation
Game playing                   poker,checker,Prisoner's dilemma
Optimization                    traveling salesman, bin packing, graph coloring
Robotics                           trajectory planning
Control                             missile evasion, pole balancing

G]Example:- To show the use of GA I have used the Six-hump Camel Back function. I will try to find the global minima of this function using GA. This function have total six minima, two of them are global. The function is: $\displaystyle f_{m2}(\vec x)=(4-2.1x_1^2+\frac{x_1^4}{3})x_1^2+x_1x_2+(-4+4x_2^2)x_2^2
with -1.9<=x1<=1.9 and -1.1<=x2<=1.1
This is the 3-D plot of the function:
and this is the contour plot of the function showing 4 local minima and 2 global minima:

I will use a package "genalg" in language R in this purpose.  This package provides a function "rbga" to implement floating point GA. This function requires a user specified evaluation function. Here is my evaluation function:

    returnValue <-  (4-(2.1*string[1]^2)+(string[1]^4/3)) * string[1]^2 + (string[1] * string[2] ) + ( -4 + (4 * string[2]^2)) * string[2]^2;

and here is the function call to GA:
and result can be checked as:
summary (rbga.results, T)

GA Settings
  Type                  = floats chromosome
  Population size       = 50
  Number of Generations = 2000
  Elitism               = 10
  Mutation Chance       = 0.1

Search Domain
  Var 1 = [-1.9,1.9]
  Var 2 = [-1.1,1.1]

GA Results
  Best Solution : -0.0899352302238796 0.712727474793792 
This result is close to one of the global minima: x1=-0.089842, x2=0.712656.
[2] “Genetic Algorithms in search, Optimization and Machine Learning” by David E. Goldberg
[3] “Neural Networks,Fuzzy Logic and Genetic Algorithms--synthesis and applications” by S.Rajasekaran and G.A. Vijayalakshmi Pai.

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